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Alternative versions of the Johnson homomorphisms and the LMO functor
Anderson Vera
To cite this version:
Anderson Vera. Alternative versions of the Johnson homomorphisms and the LMO functor. 2019.
�hal-02050022�
AND THE LMO FUNCTOR
ANDERSON VERA
Abstract. Let Σ be a compact connected oriented surface with one boundary com- ponent and letMdenote the mapping class group of Σ. By considering the action ofMon the fundamental group of Σ it is possible to define different filtrations ofM together with some homomorphisms on each term of the filtration. The aim of this paper is twofold. Firstly we study a filtration ofMintroduced recently by Habiro and Massuyeau, whose definition involves a handlebody bounded by Σ. We shall call it the
“alternative Johnson filtration”, and the corresponding homomorphisms are referred to as“alternative Johnson homomorphisms”. We provide a comparison between the alternative Johnson filtration and two previously known filtrations: the original John- son filtration and the Johnson-Levine filtration. Secondly, we study the relationship between the alternative Johnson homomorphisms and the functorial extension of the Le-Murakami-Ohtsuki invariant of 3-manifolds. We prove that these homomorphisms can be read in the tree reduction of the LMO functor. In particular, this provides a new reading grid for the tree reduction of the LMO functor.
Contents
1. Introduction 2
2. Spaces of Jacobi diagrams and their operations 7
2.1. Generalities 7
2.2. Operations on Jacobi diagrams 8
3. The Kontsevich integral and the LMO functor 11
3.1. Kontsevich Integral 11
3.2. The LMO functor 14
4. Johnson-type filtrations 23
4.1. Preliminaries 23
4.2. Alternative Torelli group 23
4.3. Alternative Johnson filtration 25
5. Johnson-type homomorphisms 29
5.1. Preliminaries 29
5.2. Alternative Johnson homomorphisms 31
5.3. Alternative Johnson homomorphism onL 35
5.4. Diagrammatic versions of the Johnson-type homomorphisms 42 6. Alternative Johnson homomorphisms and the LMO functor 45 6.1. The filtration on Lagrangian cobordisms induced by the alternative degree 45 6.2. First alternative Johnson homomorphism and the LMO functor 48 6.3. Higher alternative Johnson homomorphisms and the LMO functor 53
References 60
2010Mathematics Subject Classification. 57M27, 57M05, 57S05.
Key words and phrases. 3-manifolds, mapping class group, Torelli group, Johnson homomorphisms, Lagrangian mapping class group, LMO invariant, LMO functor, Kontsevich integral.
1
1. Introduction
Let Σ be a compact connected oriented surface with one boundary component and let M denote the mapping class group of Σ, that is, the group of isotopy classes of orientation-preserving self-homeomorphisms of Σ fixing the boundary pointwise. The group Mis not only an important object in the study of the topology of surfaces but also plays an important role in the study of 3-manifolds, Teichm¨uller spaces, topological quantum field theories, among other branches of mathematics.
A natural way to studyMis to analyse the way it acts on other objects. For instance, we can consider the action on the first homology groupH :=H1(Σ;Z) of Σ. This action gives rise to the so-called symplectic representation
σ:M −→Sp(H, ω),
whereω:H⊗H→Zis the intersection form of Σ. The homomorphismσ is surjective but it is far from being injective. Its kernel is known as theTorelli group of Σ, denoted by I. Hence we have the short exact sequence
(1.1) 1−→ I−−→ M⊂ −−→σ Sp(H, ω)−→1.
We can see that, in order to understand the algebraic structure ofM, the Torelli group I deserves significant attention because, in a certain way, it is the part of M that is beyond linear algebra (at least with respect to the symplectic representation).
More interestingly, we can consider the action ofMon the fundamental groupπ :=
π1(Σ,∗) for a fixed point ∗ ∈∂Σ. This way we obtain an injective homomorphism ρ:M −→Aut(π),
which is known as the Dehn-Nielsen-Baer representation and whose image is the sub- group of automorphisms ofπ that fix the homotopy class of the boundary of Σ.
Johnson-type filtrations. As stepwise approximations of ρ, we can consider the action of Mon the nilpotent quotients ofπ
ρm :M −→Aut(π/Γm+1π),
where Γ1π := π and Γm+1π := [π,Γmπ] for m ≥ 1, define the lower central series of π. This is the approach pursued by D. Johnson [19] and S. Morita [35]. This approach allows to define theJohnson filtration
(1.2) M ⊇ I =J1M ⊇J2M ⊇J3M ⊇ · · · whereJmM:= ker(ρm).
Now, there is a deep interaction between the study of 3-manifolds and that of the mapping class group. For instance through Heegaard splittings, that is, by gluing two handlebodies via an element of the mapping class group of their common boundary.
Thus, if we are interested in this interaction, it is natural to consider the surface Σ as the boundary of a handlebody V. Letι: Σ ,→ V denote the induced inclusion and let B :=H1(V;Z) andπ0 :=π1(V, ι(∗)). LetA and Abe the subgroups ker(H−→ι∗ B) and ker(π −→ι# π0), whereι∗ and ι# are the induced maps by ιin homology and homotopy, respectively. The Lagrangian mapping class group of Σ is the group
L={f ∈ M | f∗(A)⊆A}.
By considering a descending series (Km)m≥1 of normal subgroups ofπ(different from the lower central series) K. Habiro and G. Massuyeau introduced in [15] a filtration of
the Lagrangian mapping class group L:
(1.3) L ⊇ Ia=J1aM ⊇J2aM ⊇J3aM ⊇ · · ·
that we call thealternative Johnson filtration. We call the first termIa :=J1aMof this filtration the alternative Torelli group. Notice that Ia is a normal subgroup of L but it is not normal in M. Roughly speaking, the group Km consists of commutators of π of weight m, where the elements of A are considered to have weight 2, for instance K1 =π, K2 =A·Γ2π, K3 = [A, π]·Γ3π and so on. The alternative Johnson filtration will be our main object of study in Section 4.
Besides, in [29, 31] J. Levine defined a different filtration of L by considering the lower central series of π0, and whose first term is the Lagrangian Torelli group IL = {f ∈ L | f∗|A= IdA}:
(1.4) L ⊇ IL=J1LM ⊇J2LM ⊇J3LM ⊇ · · ·
we call this filtration the Johnson-Levine filtration. The group IL is normal in L but not inM.
We refer to the Johnson filtration, the alternative Johnson filtration and the Johnson- Levine filtration asJohnson-type filtrations. Notice that unlike the Johnson filtration the alternative Johnson filtration takes into account a handlebody. Besides, the intersection of all terms in the alternative Johnson filtration is the identity of Mas in the case of the Johnson filtration. But this is not the case for the Johnson-Levine filtration. One of the main purposes of this paper is the study of the alternative Johnson filtration and its relation with the other two filtrations. Proposition4.9and Proposition4.13give the following result.
Theorem A. The alternative Johnson filtration satisfies the following properties.
(i) T
m≥1JmaM={IdΣ}.
(ii) For all k ≥ 1 the group JkaM is residually nilpotent, that is, T
mΓmJkaM = {IdΣ}.
Besides, for everym≥1, we have
(iii) J2ma M ⊆JmM. (iv) JmM ⊆Jm−1a M. (v) JmaM ⊆Jm+1L M.
In particular, the Johnson filtration and the alternative Johnson filtration are cofinal.
Johnson-type homomorphisms. Each term of the Johnson-type filtrations comes with a homomorphism whose kernel is the next subgroup in the filtration. We refer to these homomorphisms asJohnson-type homomorphisms. TheJohnson homomorphisms are important tools to understand the structure of the Torelli group and the topology of homology 3-spheres [21,33,34, 36]. Let us review the target spaces of these homo- morphisms. For an abelian groupG, we denote byLie(G) =L
m≥1Liem(G) the graded Lie algebra freely generated by Gin degree 1.
The m-th Johnson homomorphism τm is defined on JmM and it takes values in the group Derm(Lie(H)) of degree m derivations of Lie(H). Consider the element Ω ∈ Lie2(H) determined by the intersection form ω : H ⊗H → Z. A symplectic derivation dofLie(H) is a derivation satisfying d(Ω) = 0. S. Morita shows in [35] that for h ∈ JmM, the morphism τm(h) defines a symplectic derivation of Lie(H). The group of symplectic degree m derivations of Lie(H) can be canonically identified with the kernel Dm(H) of the Lie bracket [ , ] :H⊗Liem+1(H) → Liem+2(H). This way, form≥1 we have homomorphisms
τm:JmM −→Dm(H).
The m-th Johnson-Levine homomorphism τmL :JmLM →Dm(B) is defined onJmLM and it takes values in the kernel Dm(B) of the Lie bracket [ , ] : B ⊗Liem+1(B) → Liem+2(B).
For the alternative Johnson homomorphisms [15], consider the graded Lie algebra Lie(B;A) freely generated by B in degree 1 and A in degree 2. The m-th alternative Johnson homomorphismτma :JmaM →Derm(Lie(B;A)) is defined onJmaMand it takes values in the group Derm(Lie(B;A)) of degreem derivations of Lie(B;A). Similarly to the case of Lie(H), we define a notion of symplectic derivation of Lie(B;A) by consid- ering the element Ω0 ∈Lie3(B;A) defined by the intersection form of the handlebody V. Theorem 5.9and Proposition 5.11 give the following result.
Theorem B.Let m≥1 and h∈JmaM. Then
(i) The morphism τma(h) defines a degree m symplectic derivation ofLie(B;A).
(ii) The morphism τm+1L (h) is determined by the morphism τma(h).
Property (ii) in Theorem B can be expressed more precisely by the commutativity of the diagram
JmaM ⊂ //
τma
Jm+1L M
τm+1L
Dm(B;A) ι∗ //Dm+1(B),
for m ≥ 1, where the inclusion JmaM ⊆ Jm+1L M is assured by Theorem A (v). The homomorphismι∗:Dm(B;A)→Dm+1(B) is induced by the mapι∗ :H→B. Property (i) in Theorem B allows to define a diagrammatic version of the alternative Johnson homomorphisms so that we are able to study their relation to theLMO functor. This is the second main purpose of this paper. Before we proceed with a description of our results in this setting, let us state another result in the context of the alternative Johnson homomorphisms. In [15], K. Habiro and G. Massuyeau consider a group homomorphism τ0a :L → Aut(Lie(B;A)), which we call the 0-th alternative Johnson homomorphism, and whose kernel is the alternative Torelli group Ia. In subsection 5.3 we prove the following.
Theorem C. The homomorphism τ0a : L → Aut(Lie(B;A)) can be equivalently de- scribed as a group homomorphism τ0a : L −→ Aut(B)nHom(A,Λ2B) for a certain action of Aut(B) on Hom(A,Λ2B). The kernel of τ0a is the second term J2LM of the Johnson-Levine filtration. In particular we haveIa=J1aM=J2LM.
Moreover, we explicitly describe the imageG :=τ0a(L) and then we obtain the short exact sequence
(1.5) 1−→ Ia−−→ L⊂ τ
a
−−→ G −→0 1.
This short exact sequence has a similar role, in the context of the alternative Johnson homomorphisms, to that of the short exact sequence (1.1) in the context of the Johnson homomorphisms. This is because in [15] the authors prove that the alternative Johnson homomorphisms satisfy an equivariant property with respect to the homomorphismτ0a, which is the analogue of the Sp-equivariant property of the Johnson homomorphisms.
Hence the short exact sequence (1.5) can be important for a further development of the study of the alternative Johnson filtration.
Relation with the LMO functor. After the discovery of the Jones polynomial and the advent of many new invariants, the so-calledquantum invariants, of links and
3-manifolds, it became necessary to “organize” these invariants. The theory of finite- type (Vassiliev-Goussarov) invariants in the case of links and the theory of finite-type (Goussarov-Habiro) invariants in the case of 3-manifolds, provide an efficient way to do this task. An important success was achieved with the introduction of the Kontsevich integral for links [22, 1] and the Le-Murakami-Othsuki invariant for 3-manifolds [24], because they areuniversal among rational finite-type invariants. Roughly speaking, this property says that everyQ-valued finite-type invariant is determined by the Kontsevich integral in the case of links or by the LMO invariant in the case of homology 3-spheres.
The LMO invariant was extended to a TQFT (Topological quantum field theory) in [37,6,5]. We follow the work of D. Cheptea, K. Habiro and G. Massuyeau in [5], where they extend the LMO invariant to a functor Ze:LCobq →tsA, called the LMO functor, from the category ofLagrangian cobordisms (cobordisms satisfying a homological condi- tion) between bordered surfaces to a category of Jacobi diagrams (uni-trivalent graphs up to some relations). See Figure 1.1 for some examples of Jacobi diagrams. There is still a lack of understanding of the topological information encoded by the LMO functor.
One reason for this is that the construction of the LMO functor takes several steps and also uses several combinatorial operations on the space of Jacobi diagrams. This moti- vates the search of topological interpretations of some reductions of the LMO functor through known invariants, some results in this direction were obtained in [5, 32, 43].
The second main purpose of this paper is to give a topological interpretation of the tree reduction of the LMO functor through the alternative Johnson homomorphisms.
Ahomology cobordism of Σ is a homeomorphism class of pairs (M, m) whereM is a compact oriented 3-manifold andm:∂(Σ×[−1,1])→∂M is an orientation-preserving homeomorphism such that thetop andbottom restrictionsm±|Σ×{±1}: Σ× {±1} →M ofminduce isomorphisms in homology. Denote byCthe monoid of homology cobordims of Σ (or Cg,1 where g is the genus of Σ). In particular, if h ∈ M, we can consider the homology cobordism c(h) := (Σ ×[−1,1], mh) where mh is such that mh+ = h and mh− = IdΣ. Moreover, h ∈ L if and only if the cobordism c(h) is a Lagrangian cobordism. Thusc(h) belongs to the source category of the LMO functor and therefore we can computeZe(c(h)).
The alternative Johnson homomorphisms motivate the definition of the alternative degree, denoteda-deg, for connected tree-like Jacobi diagrams. IfT is a tree-like Jacobi diagram colored by B⊕A, then
a-deg(T) = 2|TA|+|TB| −3,
where|TA|(respectively|TB|) denotes the number of univalent vertices ofT colored by A (respectively byB). See Figure 1.1(a) and (b) for some examples.
Figure 1.1. Tree-like Jacobi diagrams ofa-deg = 3 in (a) and ofa-deg = 1 in (b). (c) Looped Jacobi diagram. Herea, a0 ∈Aand b, b0, b1, . . . , b4∈ B.
Denote byTmY,a(B⊕A) the space generated by tree-like Jacobi diagrams colored by B⊕Awith at least one trivalent vertex and witha-deg =m. For a Lagrangian cobordism M letZet(M) denote the reduction ofZ(M) moduloe looped diagrams, that is, diagrams with a non-contractible connected component. See Figure 1.1 (c) for an example of a looped diagram. This way, Zet(M) consists only of tree-like Jacobi diagrams. The first step to relate the alternative Johnson homomorphisms with the LMO functor is given in Theorem6.5where we prove the following.
Theorem D.The alternative degree induces a filtration{FmaC}m≥1ofC by submonoids.
Consider the map
ZemY,a :FmaC −→ TmY,a(B⊕A),
where ZemY,a(M) is defined as the Jacobi diagrams with at least one trivalent vertex and of a-deg =m in Zet(M) for M ∈ FmaC. Then ZemY,a is a monoid homomorphism.
In Theorem6.14 and Theorem 6.16we prove the following.
Theorem E.Let m≥1and f ∈JmaM. Then the m-th alternative Johnson homomor- phism can be read in the tree-reduction of the LMO functor.
More precisely, we prove that forh∈JmaMwithm≥2, the valueZemY,a(c(h)) coincides (up to a sign) with the diagrammatic version of τma(h). For h ∈ J1aM, we show that τ1a(h) is given by Ze1Y,a(c(h)) together with the diagrams without trivalent vertices in Z(c(h)) ofe a-deg= 1. The techniques for the proof of Theorem E in the case m = 1 (Theorem6.14) andm≥2 (Theorem6.16) are different. Form= 1 we need to do some explicit computations of the LMO functor and a comparison between the first alternative Johnson homomorphism and the first Johnson homomorphism. Form≥2, the key point is the fact that the LMO functor defines an alternative symplectic expansion of π. To show this, we use a result of Massuyeau [32] where he proves that the LMO functor defines a symplectic expansion ofπ.
Theorem D and Theorem E provide a new reading grid of the tree reduction of the LMO functor by the alternative degree. Theorem E follows the same spirit of a result of D. Cheptea, K. Habiro and G. Massuyeau in [5] and of the author in [43] where they prove that the Johnson homomorphisms and the Johnson-Levine homormophisms, respectively, can be read in the tree-reduction of the LMO functor.
Notice that Theorem D holds in the context of homology cobordisms, as do the results that we use to prove Theorem E. This suggests that the alernative Johnson homomor- phisms and Theorem E could be generalized to the setting of homology cobordisms, but we have not explored this issue so far.
The organization of the paper is as follows. In Section 2we review the definition of several spaces of Jacobi diagrams and some operations on them as well as some explicit computations. Section 3 deals with the Kontsevich integral and the LMO functor, in particular we do some explicit computations that are needed in the following sections.
Section4and Section5provide a detailed exposition of the alternative Johnson filtration and the alternative Johnson homomorphisms, in particular we prove Theorem A, B and C. Finally, Section 6 is devoted to the topological interpretation of the LMO functor through the alternative Johnson homomorphisms, in particular we prove Theorem D and E.
Reading guide. The reader more interested in the mapping class group could skip Section2and Section3 and go directly to Section4 and Section5(skipping subsection 5.4) referring to the previous sections only when needed. The reader familiar with the
LMO functor and more interested in the topological interpretation of its tree reduction through the alternative Johnson homomorphisms can go directly to Section3. Then go to subsection4.3and subsection 5.2to the necessary definitions to read Section6.
Notations and conventions. All subscripts appearing in this work are non-negative integers. When we write m ≥ 0 or m ≥1 we always mean that m is an integer. We use the blackboard framing convention on all drawings of knotted objects. We usually abbreviate simple closed curve as scc. By a Dehn twist we mean a left-handed Dehn twist.
Acknowledgements. I am deeply grateful to my advisor Gw´ena¨el Massuyeau for his encouragement, helpful advice and careful reading. I thank sincerely Takuya Sakasai for helpful and stimulating discussions, in particular for explaining to me Remark4.15.
2. Spaces of Jacobi diagrams and their operations
In this section we review several spaces of diagrams which are the target spaces of the Kontsevich integral, LMO functor and Jonhson-type homomorphisms. We refer to [1,38] for a detailed discussion on the subject. Throughout this section letX denote a compact oriented 1-manifold (possibly empty) whose connected components are ordered and letC denote a finite set (possibly empty).
2.1. Generalities. Avertex-oriented unitrivalent graph is a finite graph whose vertices are univalent (legs) or trivalent, and such that for each trivalent vertex the set of half- edges incident to it is cyclically ordered.
AJacobi diagram on (X, C) is a vertex-oriented unitrivalent graph whose legs are ei- ther embedded in the interior ofX or are colored by theQ-vector space generated byC.
Two Jacobi diagrams are considered to be the same if there is an orientation-preserving homeomorphism between them respecting the order of the connected components, the vertex orientation of the trivalent vertices and the colorings of the legs. For drawings of Jacobi diagrams we use solid lines to representX, dashed lines to represent the unitriva- lent graph and we assume that the orientation of trivalent vertices is counterclockwise.
See Figure 2.1for some examples.
Figure 2.1. Jacobi diagrams with X = in (a), X = ↓ ↓ in (b) and X=∅in (c). Here C={1,2,3}.
Thespace of Jacobi diagrams on (X, C) is theQ-vector space:
A(X, C) = VectQ{Jacobi diagrams on (X, C)}
STU, AS, IHX,Q-multilinearity ,
where the relations STU, AS, IHX are local and the multilinearity relation applies to the colored legs. See Figure2.2.
Figure 2.2. Relations on Jacobi diagrams.
IfX is not empty, it is well known that, for diagramsD∈ A(X, C) such that every connected component ofDhas at least one leg attached toX, the STU relation implies the AS and IHX relations, see [1, Theorem 6]. We can also define the space A(X, G) for any finitely generated free abelian groupG asA(X, G) =A(X, C), where C is any finite set of free generators of G. If X or C is empty we drop it from the notation.
For D ∈ A(X, C) we define the internal degree, the external degree and total degree;
denoted i-deg(D), e-deg(D) and deg(D) respectively, as
i-deg(D) := number of trivalent vertices ofD, e-deg(D) := number of legs ofD,
deg(D) := 1
2(i-deg(D) + e-deg(D)).
This way, the spaceA(X, C) is graded with the total degree. We still denote byA(X, C) its degree completion.
Example 2.1. A connected Jacobi diagram inA(C) without trivalent vertices is called a strut. See Figure 2.3(a). For a matrix Λ = (lij) with entries indexed by a finite set C, we define the element [Λ] inA(C) by
Example 2.2. For a positive integern, denote bybne∗ the set{1∗, . . . , , n∗}, where∗is one of the symbols +,−or∗ itself. For instance the morphisms in the target category of the LMO functor are subspaces of the spaces A(bge+t bfe−) for g and f positive integers. See Figure2.3 (b).
Example 2.3. A Jacobi diagram inA(C) islooped if it has a non-contractible compo- nent, see Figure2.3 (b). The space oftree-like Jacobi diagrams colored byC, denoted byAt(C), is the quotient ofA(C) by the subspace generated by looped diagrams. The space of connected tree-like Jacobi diagrams colored by C, denoted by At,c(C), is the subspace of At(C) spanned by connected Jacobi diagrams in At(C). For instance the spacesAt,c(G), forGsome particular abelian groups, are the target of the diagrammatic versions of the Johnson-type homomorphisms. See Figure 2.3 (c) for an example of a connected tree-like Jacobi digram.
2.2. Operations on Jacobi diagrams. Let us recall some operations on the spaces of Jacobi diagrams.
Hopf algebra structure. There is a product inA(C) given by disjoint union, with unit the empty diagram, and a coproduct defined by ∆(D) =P
D0⊗D00where the sum ranges over pairs of subdiagramsD0, D00 of D such thatD0tD00=D. For instance:
Figure 2.3. (a) Strut, (b) Jacobi diagram inA(b4e+t b3e−), (c) Tree- like Jacobi diagram. Herea, b, c, d∈C whereC is any finite set.
With these structuresA(C) is a co-commutative Hopf algebra with counit the linear map :A(C)→Qdefined by(∅) = 1 and(D) = 0 forD∈ A(C)\ {∅}and with antipode the linear map S :A(C) → A(C) defined byS(D) = (−1)kDD where kD denotes the number of connected components of D ∈ A(C). It follows from the definition of the coproduct that the primitive part ofA(C) is the subspaceAc(C) spanned by connected Jacobi diagrams.
Doubling and orientation-reversal operations. Suppose that we can decompose the 1-manifold X asX =X0 ↓for a chosen oriented interval component of X, here X0 can be empty. Then given a Jacobi diagram D on X0 ↓ it is possible to obtain new Jacobi diagrams ∆(D) on X ↓=X0 ↓↓ and S(D) on X0 ↑. Let us represent the Jacobi diagram Das
Then ∆(D) is defined in Figure 2.4, where we use thebox notation to denote the sum over all the possible ways of gluing the legs of D attached to the grey box to the two intervals involved in the grey box, in particular if there arek legs attached to the grey box, there will be 2k terms in the sum.
Figure 2.4. Definition of the doubling map and box notation.
Besides, the Jacobi digramS(D) is given in Figure2.5.
To sum up, we have maps
(2.1) ∆ :A(↓X0)−→ A(↓↓X0) and S :A(↓X)−→ A(↑X),
calleddoubling map andorientation reversal map, respectively. Observe that even if we use the same notation for the doubling map and the coproduct, the respective meaning can be deduced from the context.
Figure 2.5. Definition of orientation-reversal map. Here we suppose that there arek legs attached to the chosen interval.
Symmetrization map. Let us recall the diagrammatic version of the Poincar´e- Birkhoff-Witt isomorphism. We follow [1, 7] in our exposition. Let D be a Jacobi diagram on (X, C t {s}) we could glue all the s-colored legs of D to an interval ↑s (labelled by s) in order to obtain a Jacobi diagram on (X ↑s, C), i.e. there would not be anys-colored leg left. But there are many ways of doing this gluing, so we consider the arithmetic mean of all the possible ways of gluing the s-colored legs of D to the interval↑s. This way we obtain a well defined vector space isomorphism
(2.2) χs:A(X, Ct {s})−→ A(X↑s, C),
called symmetrization map. It is not difficult to show that the map (2.2) is well de- fined, but it is more laborious to show that it is bijective, see [1, Theorem 8]. If S={s1, . . . , sl}, it is possible to define, in a similar way, a vector space isomorphism
χS :A(X, CtS)−→ A(X↑S, C), where↑S=↑s1 · · · ↑sl. More precisely,χS =χsl◦ · · · ◦χs1.
Example 2.4. Fix r ∈S. Denote byH(r) the subspace of A(S) generated by Jacobi diagrams with at least one component that is looped or that possesses at least two r-colored legs. Similarly, denote by H(↑r) the subspace ofA(↑S) generated by Jacobi diagrams with at least one dashed component that is looped or that possesses at least two legs attached to↑r. Bar-Natan shows in [2, Theorem 1] that χ(H(r)) =H(↑r).
The inverse of the symmetrization map is constructed recursively. Since we will use this inverse, let us review the definition. LetDbe a Jacobi diagram on (X ↑s, C) with nlegs attached to↑s. Label these legs from 1 to nfollowing the orientation of ↑s. For a permutationς ∈Sn, there is a way of obtaining a Jacobi digram ςDon (X↑s, C) by acting on the legs. For instance ifς = (123) we have:
Theorem 2.5. [1, Theorem 8] Let n≥1 and let D be a Jacobi diagram on (X ↑s, C) with at mostl≤nlegs attached to ↑s. Denote byD˜ the Jacobi diagram on(X, Ct {s}) obtained fromD by erasing↑s and coloring with s all the legs that were attached to↑s. Set σ1(D) = ˜D and for n >1
σn(D) =
D˜ +n!1 P
ς∈Snσn−1(D−ςD), if l=n, σn−1(D), if l < n.
Then the map
σ :A(X ↑s, C)−→ A(X, Ct {s}),
defined by σ(D) = σn(D) is well-defined and it is the inverse of the symmetrization map.
Example 2.6.
Example 2.7.
Example 2.8.
In the last equality we used Example2.7.
Example 2.9. We are usually interested in the reduction modulo looped diagrams. We use the symbol ≡to indicate an equality modulo looped diagrams. Using the previous examples, it is possible to show
Here the square brackets stand for an exponential, more precisely
3. The Kontsevich integral and the LMO functor
In this section we review the combinatorial definition of the Kontsevich integral from [38,25]. We also recall the construction of the LMO functor following [5]. We focus on particular examples, which will play an important role in the next sections, rather that in a detailed exposition on the subject.
3.1. Kontsevich Integral. Let us start by recalling some basic notions. Consider the cube [−1,1]3 ⊆R3 with coordinates (x, y, z). A framed tangle in [−1,1]3 is a compact oriented framed 1-manifold T properly embedded in [−1,1]3 such that the boundary
∂T (the endpoints of T) is uniformly distributed along {0} ×[−1,1]× {±1} and the framing on the endpoints of T is the vector (0,1,0). We draw diagrams of framed tangles using the blackboard framing convention. LetT be a framed tangle. Denote by
∂tT the endpoints ofT lying in{0} ×[−1,1]× {+1}, we call∂tT the top boundary of T. Similarly,∂bT of T denotes the bottom boundary.
We can associate wordswt(T) andwb(T) on{+,−}to∂tT and∂bT as follows. To an endpoint of T we associate + if the orientation ofT goes downwards at that endpoint, and − if the orientation of T goes upwards at that endpoint. The words wt(T) and wb(T) are obtained by reading the corresponding signs in the positive direction of the y coordinate. See Figure (3.1) (a) for an example of a tangle with its corresponding words.
We considernon-associative words on{+,−}, that is, words on{+,−}together with a parenthesization (formally an element of the free magma generated by {+,−}). For instance ((+−)+) and (+(−+)) are the two possible non-associative words obtained from the word +−+. From now on we omit the outer parentheses. A q-tangle is a framed tangle whose top and bottom words are endowed with a parenthesization. See Figure (3.1) (b) and (c) for two different parenthesizations of the same framed tangle.
Figure 3.1. A framed tangle and two differentq-tangles obtained from it.
To define the Kontsevich integral it is necessary to fix a particular element Φ∈ A(↓↓↓) called anassociator. The element Φ is an exponential series of Jacobi diagrams satisfying several conditions, among these, one “pentagon” and two “hexagon” equations; see [38, (6.11)–(6.13)]. From now on we fix an even Drinfeld associator Φ. In low degree we have:
Here 1 means↓↓↓. The Kontsevich integral is defined so that:
(3.1) Z(T1◦T2) =Z(T1)◦Z(T2), Z(T1⊗T2) =Z(T1)⊗Z(T2);
where the composite Z(T1)◦Z(T2) and the tensor product Z(T1)⊗Z(T2) are defined by vertical and horizontal juxtaposition of Jacobi diagrams. Now everyq-tangle can be expressed as the composition of tensor products of some elementary q-tangles, so it is enough to define the Kontsevich integral on theseq-tangles. Set
whereS2 is the orientation-reversal map applied to the second interval. The Kontsevich integral is defined on the elementaryq-tangles as follows:
and for elementaryq-tangles of the form
where the thick lines represent a trivial tangle and the black dots some non-associative words on{+,−}, the Kontsevich integral is defined by using the doubling and orienta- tion reversal maps, see subsection 2.2, for instance
Here the subscripts indicate the interval to which the operation is applied. It is known thatZ is well defined and is an isotopy invariant ofq-tangles, see [26,27]. For aq-tangle T, we denote by Zt(T) the reduction of Z(T) modulo looped diagrams, see Example 2.3.
Example 3.1.
Example 3.2.
Example 3.3. Using Examples 3.1,3.2and Equations (3.1) we have
Example 3.4. Recall the spaceH(↑r) defined in Example2.4. We have
3.2. The LMO functor. This subsection is devoted to a brief description of the LMO functor Ze :LCobq → tsA and principally to explicit computations which will be useful in the following sections. We refer to [5] for more details. Throughout this subsection we denote by Σg,1 a compact connected oriented surface of genusg with one boundary component for each non-negative integerg, see Figure3.2.
Homology cobordisms and their bottom-top tangle presentation. Let us start with some preliminaries. A homology cobordism of Σg,1 is the equivalence class of a pair M = (M, m), where M is a compact connected oriented 3-manifold and m : ∂(Σg,1×[−1,1]) → ∂M is an orientation-preserving homeomorphism, such that the bottom and top inclusions m±(·) :=m(·,±1) : Σg,1 → M induce isomorphisms in homology. Two pairs (M, m) and (M0, m0) are equivalent if there exists an orientation- preserving homeomorphism ϕ : M → M0 such that ϕ◦m = m0. The composition (M, m)◦(M0, m0) of two homology cobordisms (M, m) and (M0, m0) of Σg,1is the equiva- lence class of the pair (M , mf −∪m0+), whereMfis obtained by gluing the two 3-manifolds
M and M0 by using the map m+◦(m0−)−1. This composition is associative and has as identity element the equivalence class of the trivial cobordism (Σg,1×[−1,1],Id). De- note byCg,1 themonoid of homology cobordismsof Σg,1. This notion plays an important role in the theory of finite-type invariants as shown independently by M. Goussarov in [10] and K. Habiro in [13].
Example 3.5. Denote by Mg,1 the mapping class group of Σg,1, i.e. the group of isotopy classes of orientation-preserving homeomorphisms of Σg,1 that fix the boundary
∂Σg,1 pointwise. This group can be embedded intoCg,1 by associating to anyh∈ Mg,1 the homology cobordism, called mapping cylinder, c(h) = (Σg,1 ×[−1,1], mh), where mh :∂(Σg,1×[−1,1])→∂(Σg,1×[−1,1]) is the orientation-preserving homeomorphism defined by mh(x,1) = (h(x),1) and mh(x, t) = (x, t) for t 6= 1. This way we have an injective map c : Mg,1 → Cg,1. The submonoid c(Mg,1) is precisely the group of invertible elements of Cg,1, see [14, Proposition 2.4].
There is a more general notion of cobordism. Forg, f ≥0 letCfg denote the compact oriented 3-manifold obtained from [−1,1]3 by adding g(respectivelyf) 1-handles along [−1,1]×[−1,1]× {+1} (respectively along [−1,1]×[−1,1]× {−1}), uniformly in the y direction. A cobordism from Σg,1 to Σf,1 is the homeomorphism class relative to the boundary of a pair (M, m), where M is a compact connected oriented 3-manifold and m:∂Cfg →∂M is an orientation-preserving homeomorphism.
Given a homology cobordism (M, m) of Σg,1; or more generally a cobordism from Σg,1 to Σf,1. We can associate a particular kind of tangle whose components split in f bottom components and g top components (they are called bottom-top tangles in [5]). The association is defined as follows. First fix a system of meridians and parallels {αi, βi} on Σg,1 for each non-negative integerg as shown in Figure 3.2.
Figure 3.2. System of meridians and parallels{αi, βi}on Σg,1.
Then attach g 2-handles (or f in the case of a cobordism from Σg,1 to Σf,1) on the bottom surface of M by sending the cores of the 2-handles to the curves m−(αi). In the same way, attach g 2-handles on the top surface of M by sending the cores to the curves m+(βi). This way we obtain a compact connected oriented 3-manifold B and an orientation-preserving homeomorphism b :∂([−1,1]3) → ∂B. The pair B = (B, b) together with the cocores of the 2-handles, determine a tangle γ in B. We call the homeomorphism class relative to the boundary of the pair (B, γ), still denoted in the same way, thebottom-top tangle presentationof (M, m). Following the positive direction of they coordinate, we label the bottom components ofγ with 1−,. . .,f− and the top components with 1+,. . .,g+, respectively. This procedure is sketched in Example3.6.
Example 3.6. In Figure3.3we illustrate the procedure to obtain the bottom-top tangle presentation of the trivial cobordism Σg,1×[−1,1].
Figure 3.3. Obtaining the bottom-top tangle presentation of the trivial cobordism Σg,1×[−1,1].
Lagrangian Cobordisms. Let us now roughly describe the source categoryLCob of the LMO functor. For each non-negative integer g, let Hg = H1(Σg,1;Z) be the first homology group of Σg,1 with integer coefficients, andω:Hg⊗Hg →Zthe intersection form. Denote by Ag the subgroup of Hg generated by the homology classes of the meridians {αi}. This is a Lagrangian subgroup of Hg with respect to the intersection form. LetVg be a handlebody of genus g obtained from Σg,1 by attaching g 2-handles by sending the cores of the 2-handles to the meridians αi’s, in particular the curves αi bound pairwise disjoint disks in Vg. We also see Vg as a cobordism from Σg,1 to Σ0,1, see Figure3.4. Thus we can also seeAg asAg = ker(Hg→H1(Vg;Z)).
Figure 3.4. HandlebodyVg as a cobordism from Σg,1 to Σ0,1.
Definition 3.7. [5, Definitions 2.4 and 2.6] A cobordism (M, m) from Σg,1 to Σf,1 is said to beLagrangian if it satisfies:
• H1(M;Z) =m−,∗(Af) +m+,∗(Hg),
• m+,∗(Ag)⊆m−,∗(Af) in H1(M;Z).
Moreover, (M, m) is said to bespecial Lagrangian if it additionally satisfiesVf◦M =Vg as cobordisms.
LetM be a Lagrangian cobordism and (B, γ) its bottom-top tangle presentation. It follows, from a Mayer-Vietoris argument, thatB is ahomology cube,i.e. B has the same homology groups as the standard cube [−1,1]3, see [5, Lemma 2.12]. Notice that the
definition of q-tangle in [−1,1]3 given in subsection 3.1 extends naturally to q-tangles in homology cubes.
Let us now define the category LCob. The objects of LCob are the non-negative integers and the set of morphisms LCob(g, f) from g to f are Lagrangian cobordisms from Σg,1 to Σf,1. Denote bysLCob(g, f) the morphisms fromg tof which are special Lagrangian.
Example 3.8. Leth∈ Mg,1. Then the mapping cylinderc(h) is Lagrangian if and only ifh(Ag)⊆Ag. Moreover,c(h) is special Lagrangian if and only ifh can be extended to a self-homeomorphism of the handlebody Vg.
Let us consider some particular cases of the mapping cylinders described in Example 3.8. Letγ be a simple closed curve on Σg,1 and denote bytγ the (left) Dehn twist along γ. Recall that the mapping cylinder c(tγ) can be obtained from the trivial cobordism Σg,1×[−1,1] by performing a surgery along a (−1)-framed knot in a neighbourhood of a push-off of the curveγ in Σg,1×[−1,1], see for instance [38, Lemma 8.5]. In particular we can obtain the bottom-top tangle presentation of c(tγ) from that of Σg,1×[−1,1], see Examples 3.9,3.10and 3.11.
Example 3.9. Let tαi be the Dehn twist along a meridian curve αi. Then c(tαi) ∈
sLCob(g, g). Figure 3.5 (a) shows the bottom-top tangle presentation of the trivial cobordism Σg,1×[−1,1] (in thin line) together with a (−1)-framed knot (in thick line) such that the surgery along this knot gives the bottom-top tangle presentation ofc(tαi) showed in Figure 3.5 (b). Notice that going from Figure 3.5 (a) to Figure 3.5 (b) is exactly a Fenn-Rourke move.
Figure 3.5. Bottom-top tangle presentation of c(tαi).
Example 3.10. Letα12 be the curve shown in Figure3.6 (a) and lettα12 be the Dehn twist along α12. We have c(tα12) ∈ sLCob(g, g). As in Example 3.9, Figure 3.6 (c) shows the bottom-top tangle presentation ofc(tα12) obtained by surgery along the thick component in Figure3.6 (b).
Example 3.11. Example3.10 can be generalized. Consider two integersk and lwith 1≤k < l≤g. Let αkl be the simple closed curve which turns around the k-th handle and thel-th handle as shown in Figure 3.7(a). Consider the Dehn twisttαkl along αkl. We havec(tαkl)∈sLCob(g, g). Figure3.7 (b) shows the bottom-top tangle presentation ofc(tαkl).
Example 3.12. Let Ni be the cobordism from Σg,1 to Σg+1,1 with the bottom-top tangle presentation shown in Figure 3.8. ThenNi is a special Lagrangian cobordism.
The label r on the first (from left to right) bottom component stands for root. This is
Figure 3.6. (a) Curveα12 and (c) bottom-top tangle presentation of c(tα12).
Figure 3.7. (a) Curveαkl and (b) bottom-top tangle presentation ofc(tαkl).
Figure 3.8. Bottom-top tangle presentation of Ni ∈sLCob(g, g+ 1).
because from these cobordisms we will obtain, via the LMO functor, rooted trees with rootr that we will interpret as Lie commutators. See subsection 6.3.
Top-substantial Jacobi diagrams. Let us now describe the target category tsA of the LMO functor. The objects of the categorytsAare the non-negative integers. The set of morphisms from g tof is the subspace tsA(g, f) of diagrams in A(bge+t bfe−) (see Example2.2) without struts whose both ends are colored by elements of bge+. These kind of Jacobi diagrams are called top-substantial. If D ∈ tsA(g, f) and E ∈ tsA(h, g) the composition
D◦E=
D|j+7→j∗, E|j−7→j∗
bge∗
is the element intsA(h, f) given by the sum of Jacobi diagrams obtained by considering all the possible ways of gluing the bge+-colored legs of D with the bge−-colored legs of E. A schematic description is shown in Figure 3.9 (a). The identity morphism in
tsA(g, g) is shown in Figure 3.9(b).
Figure 3.9. (a) Composition in tsA and (b) identity morphism in tsA(g, g).
Sketch of the construction of the LMO functor. The definition of the LMO functor uses the Kontsevich integral which is defined for q-tangles. Because of this, it is necessary to modify the objects of LCob to obtain the category LCobq: instead of non-negative integers, the objects of LCobq are non-associative words in the single letter • and ifu and v are non-associative words in• of length g and f respectively, a morphism fromu tov is a Lagrangian cobordism from Σg,1 to Σf,1.
Roughly speaking, the LMO functor Ze : LCobq → tsA is defined as follows. Let M ∈ LCobq(u, v), whereu and v are two non-associative words in •. Let (B, γ0) be the bottom-top tangle presentation of M. By performing the change • 7→ (+−) in u and v we obtain words wt(γ0) andwb(γ0) on {+,−} together with some parenthesizations.
Hence γ0 is a q-tangle in the homology cube B. Next, take a surgery presentation of (B, γ0), that is, a framed link L ⊆ int([−1,1]3) and a tangle γ in [−1,1]3 such that surgery along L carries ([−1,1]3, γ) to (B, γ0). Setwt(γ) =wt(γ0) and wb(γ) =wb(γ0).
Hence L∪γ is a q-tangle in [−1,1]3. Now, consider the Kontsevich integral of L∪γ, which gives a series of a kind of Jacobi diagrams. To get rid of the ambiguity in the surgery presentation, it is necessary to use some combinatorial operations on the space of diagrams. Among these operations there is the so-calledAarhus integral (see [3,4]), which is a kind of formal Gaussian integration on the space of diagrams. We then arrive totsA. Finally, to obtain the functoriality, it is necessary to do a normalization.
Recall that the definition of the Kontsevich integral requires the choice of aDrinfeld associator, and the bottom-top tangle presentation requires the choice of a system of meridians and parallels. Thus, the LMO functor also depends on these choices.
We are especially interested in the LMO functor for special Lagrangian cobordisms.
For these kind of cobordisms the LMO functor can be computed from the Kontsevich integral and the symmetrization map as is assured by a result of Cheptea, Habiro and Massuyeau. We state the result for our particular case.
Convention 3.13. From now on, we endow Lagrangian cobordisms with the right- handed non-associative word (• · · ·(•(••))· · ·) in the letter • unless we say otherwise.
This way we will always be in the context of the categoryLCobq.
Lemma 3.14. [5, Lemma 5.5] LetM ∈ LCobq(u, v), whereuandv are non-associative words in the letter• of length g and f, respectively. Suppose that the bottom-top tangle presentation ofM is as in Figure 3.10, where T is a tangle in [−1,1]3. Endow T with the non-associative words wt(T) = u/•7→(+−) and wb(T) = v/•7→(+−). Then the value of the LMO functor Z(M)e can be computed from the value of the Kontsevich integral Z(T) as shown in Figure3.11.
Let (M, m) be a homology cobordism and (B, γ) its bottom-top tangle presentation.
Define thelinking matrix of (M, m), denoted Lk(M), as the linking matrix of the link
Figure 3.10. Bottom-top tangle presentation of M.
Figure 3.11. Value ofZ(Me ) in terms ofZ(T).
ˆ
γ inB obtained fromγ by identifying the two endpoints on each of the top and bottom components of γ.
For any Lagrangian cobordismM, denote byZes(M) the strut part ofZe(M), that is, the reduction ofZ(M) modulo diagrams with at least one trivalent vertex. Denote bye ZeY(M) the reduction ofZe(M) modulo struts. Denote byZet(M) the reduction ofZe(M) modulo looped diagrams. Finally denote by ZeY,t(M) the reduction of Zet(M) modulo struts.
Lemma 3.15. [5, Lemma 4.12] LetM ∈ LCobq(u, v)where uandvare non-associative words in the letter •. Then Z(M)e is group-like. Moreover Ze(M) = Zes(M)tZeY(M) and
(3.2) Zes(M) =
Lk(M) 2
.
The colors 1+, . . . , g+ and 1−, . . . , f− in the series of Jacobi diagramsZ(M) refer toe the curvesm+(β1),. . .,m+(βg) andm−(α1), . . . , m−(αf) on the top and bottom surfaces ofM respectively.
Example 3.16. Let us consider the special Lagrangian cobordismc(tαi), from Example 3.9, equipped with non-associative words as in Convention 3.13. By Lemma 3.14 and the functoriality ofZ(see Equation (3.1)), to computeZet(c(tαi)) in low degrees we need to first compute
in low degrees, which we already computed in Example 3.2. Therefore
From Example2.9, we conclude
which shows that there are no terms of i-deg = 1 inZeY,t(c(tαi)).
Example 3.17. Consider the special Lagrangian cobordismc(tα12) from Example3.10, equipped with non-associative words as in Convention3.13. By Lemma3.14, to compute Zet(c(tα12)) in low degrees, we need to first compute the tree-like part in the Kontsevich integral of theq-tangle
by the functoriality of Z, see (3.1), we have to compute the low degree terms of
which was computed in Example3.3. Now, by a straightforward but long computation we obtain
Example 3.18. Example3.17can be generalized to the cobordismc(tαkl) from Example 3.11. In this case we obtain
Example 3.19. Consider the special Lagrangian cobordism N1 from Example 3.12, equipped with non-associative words as in Convention 3.13. Denote by w the right- handed non-associative word in • of length g−1. Denote by P•,•,w the q-cobordism ((••)w)→(•(•w)) whose underlying cobordism is the identityLCob(g+ 1, g+ 1). Thus we can decompose N1 as N1 = P•,•,w◦(T ⊗Idw), where T is the special Lagrangian cobordism whose bottom-top tangle presentation is shown in Figure3.12.
Figure 3.12. Bottom top-tangle presentation of T.
Hence,Zet(N1) =Zet(P•,•,w)◦(Zet(T)⊗Idg−1). Now, by the functoriality of Zewe have
Z(Pe •,•,w)|r7→0
=∅⊗Idg and
Z(Pe •,•,w)|1−7→0
= Id1⊗∅⊗Idg−1, therefore
ZeY(P•,•,w)|r7→0
=∅ and
ZeY(P•,•,w)|1−7→0
=∅.
This way, each one of the connected diagrams appearing in ZeY(P•,•,w) has at least one r-colored leg and at least one 1−-colored leg. Hence, each one of the connected diagrams inZet(N1) coming fromZeY(P•,•,w) has at least oner-colored leg and at least one 1−-colored leg.
We are interested in the low degree terms ofZet(N1) modH(r). By Lemma3.14, we need to compute the low degree terms of
which we already computed in Example3.4. Whence we obtain
We conclude that each of the terms with i-deg = 1 inZ(Ne 1) mod H(r) has oner-colored and one 1−-colored leg. In a similar way, it can be shown for 1≤i≤g that each of the terms with i-deg = 1 in Z(Ne i) modH(r) has oner-colored leg and onei−-colored leg.
4. Johnson-type filtrations
As in subsection3.2, we denote by Σg,1a compact connected oriented surface of genus gwith one boundary component. LetMg,1 denote the mapping class group of Σg,1. We will often omit the subscriptsgand 1 of our notation unless there is ambiguity, then we will usually write Σ and Minstead of Σg,1 and Mg,1.
4.1. Preliminaries. Let us fix a base point ∗ ∈ ∂Σ and set π = π1(Σ,∗) and H = H1(Σ,Z), finally denote by ab : π → H the abelianization map. Notice that the intersection formω :H⊗H→Zis a symplectic form onH. The elements ofMpreserve
∂Σ, in particular they preserve∗, therefore we have a well defined group homomorphism:
(4.1) ρ:M −→Aut(π),
which sends h ∈ M to the induced map h# on π. It is well known that the map ρ is injective and it is called theDehn-Nielsen-Baer representationofM. On the other hand, since the elements of Mare orientation-preserving, their induced maps on H preserve the intersection form. This way we have a well defined surjective group homomorphism:
(4.2) σ :M −→Sp(H) ={f ∈Aut(H)| ∀x, y∈H, ω(f(x), f(y)) =ω(x, y)}, that sends h ∈ M to the induced map h∗ on H. The map σ is called the symplectic representation ofMand it is far from being injective, its kernel is known as theTorelli group of Σ, which is denoted by I (or Ig,1), so
(4.3) I =Ig,1 = ker(σ) ={h∈ M |h∗= IdH}.
4.2. Alternative Torelli group. LetV (orVg) be a handlebody of genusg. Consider a diskDon∂V such that∂V = Σ∪D, whereDand Σ are glued along their boundaries.
Letι: Σ,→V be the inclusion of Σ into ∂V ⊆V, see Figure4.1.
Figure 4.1. The inclusion Σ,−→ι V.
Figure 4.1 also shows the fixed system of meridians and parallels of Σ used in sub- section 3.2. Moreover we suppose that the images ι(αi) of the meridians αi, under the embeddingι, bound pairwise disjoint disks inV. SetH0 =H1(V;Z) andπ0=π1(V, ι(∗)) and denote by ab0 :π0 →H0 the abelianization map. Consider the following subgroups of π and H that arise when looking at the induced maps by ι in homotopy and in homology:
(4.4) A= ker(ι∗ :H →H0) and A= ker(ι#:π →π0).
We also consider the following subgroup of π:
(4.5) K2= ker(π−→ι# π0 −→ab’ H0) =A·Γ2π.
The subgroup A ≤H is a Lagrangian subgroup of H with respect to the intersection form onH and it is the group that appears in the definition of Lagrangian cobordisms in the previous section. We may think of K2 as the subgroup of π generated by com- mutators ofweight 2, where the elements ofπ are considered to have weight 1 and the elements ofAare considered to have weight 2. The subgroups A,AandK2 allow us to define some important subgroups of the mapping class group M.
Definition 4.1. The Lagrangian mapping class group of Σ, denoted by L (orLg,1) is defined as follows:
(4.6) L=Lg,1={f ∈ Mg,1 | f∗(A)⊆A}.
We are mainly interested in three particular subgroups ofL, one of these is the Torelli group, see equation (4.3).
Definition 4.2. The Lagrangian Torelli groupof Σ, denoted byIL(orIg,1L ), is defined as follows:
(4.7) IL=Ig,1L ={h∈ L | h∗|A= IdA}.
The groupsLandILappear in the works [29,31] of J. Levine in connection with the theory of finite-type invariants of homology 3-spheres. From an algebraic point of view these groups were studied by S. Hirose in [16], where he found a generating system for L and by T. Sakasai in [41], where he computedH1(L;Z) and H1(IL;Z).
Definition 4.3. The alternative Torelli group of Σ, denoted by Ia (or Ig,1a ), is defined as follows:
(4.8) Ia =Ig,1a =
forx∈π: h#(x)x−1∈K2 h∈ L and for y∈K2 :
h#(y)y−1 ∈Γ3π·[π,A] =:K3
.
Notice that the definition of Ia involves the group K3 = Γ3π ·[π,A] = [[π, π], π]· [π,A], which we see as the subgroup of π generated by commutators of weight 3. Like the Lagrangian Torelli group, the group Ia appears in [29, 31, 9] in connection with the theory of finite-type invariants but with a different definition: the second term of the Johnson-Levine filtration. Definition 4.3 comes from [15], see Proposition for the equivalence of the two definitions. J. Levine shows in [29, Proposition 4.1] that Ia is generated by Dehn twists along simple closed curves (scc) whose homology class belongs toA. Equivalently, Ia is generated by Dehn twists along scc’s which bound a surface in the handlebody V. This is the definition ofIa given in [31,9].
From the above definitions it follows thatI ⊆ IL⊆ Land Ia⊆ IL⊆ L. ButIa6⊆ I andI 6⊆ Ia. We shall call here the groupsI,IL andIa Torelli-type groups. In contrast withI, the groupsIL and Ia are not normal inM, but they are normal in L.
Example 4.4. The Dehn twiststαi and tαkl from Examples3.9and 3.11are elements of the alternative Torelli group which do not belong to the Torelli group.
Example 4.5. Consider the parallelβ1 and the curveγ as shown in Figure 4.2. These curves form a bounding pair. Consider the Dehn twists tβ1 and tγ along these curves.
It can be shown that the homeomorphismtγt−1β
1 belongs to I ∩ Ia.
Figure 4.2. Curves β1 and γ.
More generally we have the following lattice of subgroups:
Ia r
$$
I ∩ Ia +
88
s
&&
IL //L
I ,
::
where all the inclusions are proper. Besides, J. Levine proved in [28, Theorem 2] that
(4.9) I ∩ Ia=K ·[I,Ia],
whereK is theJohnson kernel. D. Johnson proved in [20] thatKis generated by BSCC maps (bounding scc’s), that is, Dehn twists along scc’s which are null-homologous in Σ.
4.3. Alternative Johnson filtration. This subsection is devoted to the study of a filtration of the alternative Torelli group introduced in [15] which we shall call here the alternative Johnson filtration. We compare this filtration with the Johnson filtration and theJohnson-Levine filtration. Let us start by recalling some terminology.
An N-series (Gm)m≥1 of a groupG is a decreasing sequence G=G1 ≥G2 ≥ · · · ≥Gm≥Gm+1≥ · · ·
of subgroups ofG such that [Gi, Gj]⊆Gi+j fori, j ≥1. We are interested in N-series of the groupπ=π(Σ,∗). A first example of an N-series ofπ is the lower central series (Γkπ)k≥1. We consider an N-series of π in which the subgroupAplays a special role.
Set K1 = π and K2 = A·Γ2π as defined in Equation (4.5). Let (Km)m≥1 be the smallestN-series ofπ starting with theseK1andK2, that is, if (Gi)m≥1 is anyN-series of π with G1 =K1 and G2 =K2 then Km ⊆Gm for every m ≥1. More precisely, for everym≥3 we have
(4.10) Km = [Km−1, K1]·[Km−2, K2].
In particularK3 = Γ3π·[π,A] is the group that we used in the definition of the alter- native Torelli group, see (4.8). We can think ofKm as the subgroup ofπ generated by commutators of weightm, where the elements ofπ have weight 1 and the elements of Ahave weight 2. By induction on m≥1 we have
(4.11) Γmπ ⊆Km⊆Γdm/2eπ,
wheredm/2e denotes the least integer greater than or equal to m/2.
Restricting the Dehn-Nielsen-Baer representation (4.1) to the Lagrangian mapping class group we get an action of Lon K1 =π. We denote the action of h∈ Lon x ∈π by hx. Hencehx=ρ(h)(x) =h#(x).
Lemma 4.6. For every h∈ L we have h(K2) =K2.
